How to find exponential objects and subobject classifiers in a given category

In a course I'm learning about Topos theory, there are a lot of exercises which require you to prove explicitly some category is an elementary topos: i.e. to construct exponentials and a subobject classifier, and to show that it has all finite limits. Despite having worked my way through a number of these questions, I still find them very difficult and have not yet found a consistent method of approaching them. Examples include:

• The functor category $[\mathcal{C}^{\text{ op}}, \bf{Set}]$ for a small category $\mathcal{C}$.

• The category $B(G)$ of continuous $G$-sets with $G$-actions (for some group $G$), $(X,\epsilon)$ and as morphisms $(X,\epsilon ) \to (Y, \epsilon ')$ the functions $X \to Y$ which respect the group actions.

• The functor category $[\mathcal{C}^{\text{ op}}, \bf{Set}]$ for a category $\mathcal{C}$ with slice category $\mathcal{C}/c$ equivalent to a small category for each object $c$.

• The category of graphs which permit multiple directed edges between two vertices.

This is a soft question, since I am aware of how to do the first of these exercises (using the Yoneda lemma) and the third appears to require a little extra work and the result of the first, but in general it's not very clear to me how to go about trying to construct an exponential, or find a subobject classifier, or even necessarily to show that a category has all finite limits - you can use results like having products and equalisers to deduce having all limits, but then you still have to show the existence of certain limits, exponentials and a subobject classifier.

Intuitively speaking, is there any sensible way to go about trying to show that these things exist for a specific category? I guess you have to construct them explicitly for these sorts of exercises, but the trick with the Yoneda lemma for the top one will only work for functor categories; in general, how would you approach construction of these objects? My question is vague so I'm happy with vague answers, just trying to get better at tackling these sorts of exercises and any thoughts you can throw in would be much appreciated, thanks.

Solution 1:

Maybe this is not an answer but an example.

If $\mathcal E$ is an elementary topos, then the category $\mathcal E^G$ of (right) $G$-objects for a group object in $\mathcal E$ is again an el. topos, and the proof is pretty easy using those objects you already have in $\mathcal E$. You can find it (and more) in Mac Lane-Moerdijk.