If $F:A\to C$ and $G:B\to C$ are morphisms in $Cat$, then their pseudo-pullback (I hope it's the right notion) can be calculated as the strict limit $A\times_C C^I \times_C B$, where $I$ is the category with two objects and one isomorphism between them.

If we take $I$ to be the category with two objects and one **morphism** between them, then the strict limit above becomes the category that has:

- Objects are $(a,b,f)$, where $a\in Ob(A)$, $b\in Ob(B)$ and $f:F(a)\to G(b)$ is a morphism in $C$.
- A morphism from $(a,b,f)$ to $(a',b',f')$ is a pair $(g,h)$ such that $g:a\to a'$ is a morphism in $A$, $h:b\to b'$ is a morphism in $B$ and we have: $G(h)\circ f=f'\circ F(g)$.

My question is, what did we get? Is it also some sort of a 2 pullback?

Going one dimension higher, if $F:A\to C$ and $G:B\to C$ are morphisms in $2-Cat$ we can consider the 2 category that has:

- Objects are $(a,b,f)$, where $a\in Ob(A)$, $b\in Ob(B)$ and $f:F(a)\to G(b)$ is a 1 morphism in $C$.
- A 1 morphism from $(a,b,f)$ to $(a',b',f')$ is a triple $(g,h,\alpha)$ such that $g:a\to a'$ is a morphism in $A$, $h:b\to b'$ is a morphism in $B$ and $\alpha$ is a 2 morphism in $C$ from $f'\circ F(g)$ to $G(h)\circ f$.
- A 2 morphism from $(g,h,\alpha)$ to $(g',h',\alpha')$ is a pair $(k.l)$ such that $k:g\to g'$ is a 2 morphism in $A$, $l:h\to h'$ is a 2 morphism in $B$ and we have: $(G(l)\circ id_{f})\circ \alpha=\alpha'\circ (id_{f'}\circ F(k))$.

Again my question is, what do we get? Is it some sort of a 3 pullback?

What is a good reference for these sort of things?

I'm asking because such a construction arose naturally in my work.