Locally cartesian closed vs cartesian closed If I have a cartesian closed category (i.e. it has finite limits and exponential objects), is it also locally cartesian closed (see https://ncatlab.org/nlab/show/locally+cartesian+closed+category). Intuition (and terminologies) seem to imply the following inference:
cartesian closed $\implies$ locally cartesian closed
(the converse does not hold unless we add the condition that the locally cartesian closed category contains a terminal object)
However, in the same page of ncatlab, the author of the article asserts that "There are categories which are cartesian closed and not locally cartesian closed" (an example $Cat$). Well, my question is the follow why it has been chosen to use the term locally cartesian closed in order to describe a condition which is not weaker than cartesian closedness?
 A: It's not always the case that being blah immediately means you're locally blah. For a simple example from topology (which I assume is the origin of the "local" terminology), the cone on the hawaiian earring is simply connected, but is not locally simply connected.
So in general, there's no relationship between being blah and being locally blah. Of course, in practice it's often the case that local blah-ness is weaker than global blah-ness, which is what leads to this kind of (extremely common!) misunderstanding.
So what's the difference between locally cartesian closedness and regular cartesian closedness?
Being locally cartesian closed tells you that each slice category is cartesian closed. This is "local" in the sense that a slice category $\mathcal{C} / x$ is the part of the category "near" $x$. For instance, if you restrict attention to the monic arrows into $x$, you get exactly the subobjects of $x$.
As a source of intuition, knowing that $\mathcal{C}$ is cartesian closed tells us that "functions spaces" exist. But it's not clear why the existence of function objects in $\mathcal{C}$ would have anything at all to do with each object having a heyting algebra of subobjects.
As a source of concrete examples, the situation is a bit more delicate (or at least, the only examples I know of are a bit delicate. I would love to see someone provide easier ones in another answer, or in the comments). You can find an example of a cartesian closed but not locally cartesian closed category here. Conversely, you can find an example of a locally cartesian closed category that isn't cartesian closed here.

I hope this helps ^_^
