A problem in category theory:
Suppose $F : \mathcal C \to \mathcal D$ has a right adjoint. Fix a functor $t: I \to \mathcal C$ with $I$ small, and suppose that $t$ has a colimit. Prove that $\lim_{\to} (F \circ t)$ exists and satisfies $\lim_{\to} (F \circ t) \cong F(\lim_{\to} t)$.
I think we are to use the adjoint functor, but I have yet to find a solution for this. Any help would be great.
This is the claim that left adjoint functors preserve all colimits that exist in the domain category. You should be able to find proofs online or in textbooks. Without further details of your attempt and where you got stuck, I'd suggest you start at the beginning: Suppose that a colimit with a universal cone is given for $t$ in $\mathcal C$. You want to verify the universal property for $F$ applied to this universal cone. So, take any cone in $\mathcal D$. The right adjoint allows you to move it from $\mathcal D$ back to $\mathcal C$ where you can compare it to the universal cone there. Then you can carry the information that will give you along $F$ to $\mathcal D$ again. This is a very straightforward approach to proving this result, and, with some care, it works and it is informative since it shows you what is going on.