Q:

name this "hybrid" categorical construction

I've found a general categorical construction which I'm not familiar with.
Suppose that we have the square shown, with categories $A$, $B_i$, $C$ and functors $F_i$ and $G_i$ such that the diagram commutes ($G_0 F_0 = G_1 F_1$), and that $A$ is discrete in the sense that it has only the identity morphisms.
$$
\begin{array}{rcccl}
A & {} & \xrightarrow{F_0} & {} & B_0 \\
{} & {} & {} & {} & {} \\
{\scriptstyle F_1}\downarrow & {} & {} & {} & \downarrow{\scriptstyle G_0} \\
{} & {}  & {} & {} & {} \\
B_1 & {} & \xrightarrow{G_1} & {} & C \\
\end{array}
$$
Then define $A^*$ as follows: the objects of $A^*$ are the same as the objects of $A$; a morphism $m \in A^*(X, Y)$ is a pair $(m_0, m_1)$ such that $m_i \in B_i(F_i(X), F_i(Y))$ and $G_0(m_0) = G_1(m_1)$.  I believe that this makes $A^*$ a category with obvious functors $A \to A^*$, $A^* \to B_i$ making the whole diagram commute.
$$
\begin{array}{rcccl}
A & {} & \xrightarrow{F_0} & {} & B_0 \\
{} & \searrow  & {} & \nearrow & {} \\
{\scriptstyle F_1}\downarrow & {} & A^* & {} & \downarrow{\scriptstyle G_0} \\
{} & \swarrow  & {} & {} & {} \\
B_1 & {} & \xrightarrow{G_1} & {} & C \\
\end{array}
$$
For example, suppose $A$ is the discrete category $\operatorname{Top}_{\operatorname{dis}}$ of topological spaces with identity morphisms, $B_0$ is the usual category $\operatorname{Top}$, $B_1$ is the category $\operatorname{Set}_{\operatorname{inj}}$ of sets with injective maps between them, and $C$ is the usual category $\operatorname{Set}$.  The functors are all the obvious inclusions/forgettings.  Then $A^*$ is the category of topological spaces with injective continuous maps between them.  It's a kind of hybrid between $\operatorname{Top}$ and $\operatorname{Set}_{\operatorname{inj}}$.
$$
\begin{array}{rcccl}
\operatorname{Top}_{\operatorname{dis}} & {} & \xrightarrow{} & {} & \operatorname{Top} \\
{} & \searrow  & {} & \nearrow & {} \\
{}\downarrow & {} & A^* & {} & \downarrow{} \\
{} & \swarrow  & {} & {} & {} \\
\operatorname{Set}_{\operatorname{inj}} & {} & \xrightarrow{} & {} & \operatorname{Set} \\
\end{array}
$$
Questions

Does this all work?
Is this a known construction?  In particular, does it have some nice universal property?

A:

$A^*$ is the functor pullback (thank you to responders above) combined with the "inverse image of a functor" described in my other question here.  We can get the pullback $P$, then its universal property gives us a functor $F: A \to P$ with a commutative diagram.  Then in the notation of that other question, we can factor $F$ into $A \to F^{-1}(P) \to P$, and take $A^* = F^{-1}(P)$.
I'm still not sure whether this is useful.  I need to think some more about the principle of equivalence invariance which responders have mentioned.

