We know that the category of presheaves (i.e. $Fct(C^{Op}, Set)$ ) is an elementary topos and in particular it is finitely complete and finitely cocomplete and so it has all finite limits and colimits.
For instance I know that the terminal object is a functor $\Bbb{1} :C^{Op}\rightarrow Set$ which sends all objects to a fixed singleton and all arrows to the identity from that sigleton to itself. I know also the binary product, the equalizer and the initial object. Can you help me to find the other remarkable limits and colimits (pullbacks, coproduct, coequalizer and pushout?)
We know also that an elementary topos is always a regular category, who are the internal equivalence relations and the kernel pairs?
Last question: I know that a mono in presheaves is a natural transformation between two functors $\eta :F\rightarrow G$ if and only if $\eta_A$ is injective for all $A\in Obj(C^{Op})$. Is it true that an arrow $\mu$ in the category of presheaves is an epi iff $\mu_A$ is surjective for all $A\in Obj(C^{Op})$ ?