Generally speaking, however, Fe just lends itself more towards empathizing with the emotions of others. Fe almost experiences the emotions of others
Not really. Fe can see the emotional state someone is in and it automatically knows the role that they must play in the scenario that has been given. This could mean to support the person due to the external standards that one is supposed to do in that situation but it could also just as easily be to ignore their dramatic outburst if the person responds unfavorably or wrongly in a certain situation.
A Fe user also doesn't necessarily feel the emotions that the person is experiencing, they are more or less just responding in a way that they know they are supposed to respond. If that is lending an ear, offering words of comfort, keeping the peace between two fighting family members they will do that. That is the way they believe they should respond in context to the situation. Fe is more concerned with adhering to the objective standard of emotional responses, it is not directly correlated with feeling directly what that person is feeling or even mirroring emotions although it can do this and can lead to feeling for the person, even with the person.
The notion or belief that Fe is
automatically empathetic is incorrect, as is that same idea for Fi. The user has to feel those feelings in order to be sympathetic or empathetic which is a different ballgame. Fe and Fi are both judging functions at their base terms. Fe judges the criteria externally (is the person acting appropriately on the guidelines that I know about? Do they deserve my actions, What action will I take in regards to this current situation, How do I deal with them, etc.?) then it acts. Fi judges the critera internally (is what they are doing right according to my personal subjective view of the situation, what is my view on this situation and is it applicable, what would I do if I was in a similar position/what would I want done for me, etc.) and then it acts. Fi is a subjective clarification process and is refined and intune with the person own personally thoughts, values, ideas, etc.
Both Fi and Fe are simply judging the content outside or inside there is no direct hard wired emotional response off the bat. Both Fe and Fi could feel sympathy or empathy for others based on their judgments. Neither is more likely to be sympathetic or empathetic then the other one. Emotional responses are based purely on the individual person using those functions. Fe could be more expressive and could be more caring, but at the same time so could Fi. I have ENFJ, ESTx, ESFP cousins and all show affection in similar ways they use various forms of either Fi or Fe. All can be caring in their own ways.
Take an ISFP and an ESFJ. Which type is more likely to show empathy? Which type is more likely to pay attention to the emotions of others? Not that an ISFP is incapable of doing these things or doing them well, but an ESFJ is more likely to do these things because they use Fe rather than Fi.
This isn't 100% true. I have seen ESFJ's who blatantly do not care at all about someones personal beliefs because they believed they were right about a course of action even though they completely undercut the autonomy of another individual they loved. Fe can be just as unemotional and unsympathetic as Fi can be. There are dark sides to all functions. Fi is not inherently less capable of empathy then Fe is. Again it all comes down to the persons who are using these functions. It's ridiculous to say that a judging function makes one incapable of having sympathy or empathy for others. That's the own users personal emotions coming into the mix. Fe and Fi are not emotions. That being said you could say that Fe is more demonstrative and more likely to express their thoughts and enforce them in a group setting because they feel supported by the group -- the groups make their values. You could also say that Fi is less likely to openly express their judgements, values and principals because those are internal for them and subjective to them. The protect them extensively. Fi is internally focused and Fe is externally focused. But when it comes down to interpersonal interactions that's largely based on the person, context and situation. Not a judgement that just tells you if something is right or bad.
Focusing on the emotions of others fosters empathy.
How so? Fe is only judging the external persons behavior in order to make a decision, just like Fi is, the only difference between this judgement process is that Fe looks at external methods (how others are responding to the person, what the person should be doing/acknowledging, how the person should be acting, circulating judgement on what the objective right action during a situation would be, etc.) while Fi looks at subjective ones (Is this person doing what I would do, are they adhering to my person criteria of good actions, do they have viable motivations/reasoning's for doing what they did according to my personal systems, etc.). Nothing about Fi or Fe is emotional. Fi and Fe are rational processes based on what they "feel" is good or bad by their own systems. This can stem towards a persons "emotionality" for a lack of a better term but at their rawest cores Fe and Fi are simply judgement functions - empathy and sympathy are later factors based on the individual's own personality or temperment. Not something adherent to a cognitive function that helps make choices in judging the world around them.
Fostering empathy leads to being more natural at empathizing.
I don't see how one is inept at fostering empathy. There is nothing that states Fi can't foster empathy or can't focus on others. Fi inherently likes personal context, authentic relationships and one-on-one style communication, all of this in theory could breed intimacy and thus empathy/emotional connection. Empathy is simply put, putting yourself in someone elses place and feeling what you would imagine they would feel -- Fi can definitely do that. If you listen to a Fi-dom discuss these things that's in actuality the very way in which they come to care about people through the concept of treating other people they imagine they would want to be treated.
I once again stress that Fe is only more natural at empathy than Fi is, Fi is certainly just as capable.
It's not though, that's just a stereotype. Empathy is more about personal ability more then anything else. Feeling in Jung's terms is not emotions.