Relation between limit on a grid and limit of the diagonal Let $\mathcal{C}$ be a category with all limits and suppose we have a functor
$$F:\mathbb{N}^2\longrightarrow\mathcal{C}\ ,$$
where $\mathbb{N}^2$ is seen as the category with morphisms generated by $(n+1,m)\to(n,m)$ and $(n,m+1)\to(n,m)$. Can anything be said about the relations between
$$\lim_{n,m}F(n,m)\qquad\text{and}\qquad\lim_k F(k,k)?$$
In particular, are there assumptions one can make so that they are naturally isomorphic? I tried drawing a couple diagrams, but I got nowhere.
 A: No assumption is needed. You have a canonical morphism $$\displaystyle\lim_{n,m} F(n,m) \to \lim_{k} F(k,k)$$ constructed as follow: take the canonical cone $(\lim_{n,m} F(n,m) \to F(k,\ell))_{k, \ell}$ and forget about the non-diagonal terms, you end up with a cone $(\lim_{n,m} F(n,m) \to F(k,k))_{k}$ that must factor through the limit $\lim_k F(k,k)$.

The claim is that this canonical map is always iso.

Not very formally, it goes as follow: to define the inverse map is the same as defining a specific cone $(\lim_k F(k,k) \to F(n,m))_{n,m}$ which is constructed at component $(n,m)$ by selecting any $p\geq \max(n,m)$ and composing $\lim_k F(k,k) \to F(p,p)$ with $F(p,p) \to F(n,m)$. You can check that this i s well defined and that it induces indeed an inverse for the canonical map above.

But you can also call the big guns because this is know in greater generality. The previous claim is equivalent to this one:

The diagonal functor $d:\mathbb N \to \mathbb N^2$ is co-cofinal. 

So all we have to check is the following property: for each $(n,m)$, the comma category $(d\downarrow (n,m))$ is non empty and connected. This is the case:


*

*for any $(n,m)$, any $p \geq \max(n,m)$ gives an object $(p,p)\to (n,m)$ of the category $(d \downarrow(n,m))$, hence the non-emptyness

*for two such objects $(p,p)\to (n,m)$ and $(q,q)\to (n,m)$, select the greater between $p$ and $q$ (say $q$ here), giving a morphism $(q,q) \to (p,p)$ of the category $(d \downarrow(n,m))$, hence the connectedness.

