What is meant by $\lim_I \lim_J F$ where $F$ is a functor?

Question

I am reading some notes on category theory and there is a theorem that says:

What exactly is meant by $\lim_I \lim_J F$? Since $\lim_J F$ is an object in $\mathcal{C}$ I am not sure what they mean here.

Answer 1

Let $F:I\times J\to\mathcal C$ be a functor.

For each object $i$ of $I$, we can consider the functor $F(i,-):J\to\mathcal C$ defined by $j\mapsto F(i,j)$ for all objects $j$ of $J$. Since $F(i,-):J\to\mathcal C$ is a functor, we can take its limit over $J$, that's $\lim_JF(i,-)$ which is an object of $\mathcal C$.

The map $i\mapsto\lim_JF(i,-)$ defines a functor $I\to\mathcal C$ which is denoted by $\lim_JF$. Thus for all objects $i$ of $I$ we have $(\lim_JF)(i)=\lim_JF(i,-)$.

Since $\lim_JF:I\to\mathcal C$ is a functor, we can finally compute the limit $\lim_I\lim_JF$ which is an object of $\mathcal C$.