Let $A, B$ be two abelian categories, and $\tau : A \to B$ a left exact functor.

We define a category $C$ as follows:

objects: triples $(M, N, \varphi)$ where $M\in A, N\in B$ and $\varphi: N\to \tau M$ is a morphism in $B$.

morphisms: pairs $(f, g): (M, N, \varphi) \to (M^{\prime}, N^{\prime}, \varphi^{\prime})$ where $f:M\to M^{\prime}$ and $g:N\to N^{\prime}$ satisfying $(\tau f) \circ \varphi = \varphi ^{\prime} \circ g$.

Then, is the following statement true?

If so, then how can one prove it?

STATEMENT: If $A, B$ have enough injectives, then so does $C$.

For example, let $X$ be a scheme, $Y$ be a closed subscheme of $X$, and $U=X\setminus Y$. If $A$=(etale sheaves on $X$), $B$=(etale sheaves on $U$), then the cagegory $C$ is equivalent to the category of etale sheaves on $Y$. So, $C$ has enough injectives , of course. I wonder whether this kind of situation happens in the general setting above.

Please give me any advice.