Category of Morphisms Between Modules

Question

Let $A$ be a connected finite dimensional basic $k$-algebra with $k$ an algebraically closed field, and denote by $mod(A)$ the category of finite dimensional left $A$-modules.

We define the category $Mor\big(mod(A)\big)$, whose objects are the morphisms in $mod(A)$, and for two such morphisms $f:A \to B$ and $g:M\to N$, a morphism from $f$ to $g$ in $Mor\big(mod(A)\big)$ will be a pair of morphisms in $mod(A)$, $(u,v):f \to g$ such that $u:A \to M$ and $v: B \to N$ such that the induced diagram commutes. The composition and identities are the obvious ones.

If $gl.dim(A)=0$, then $Mor\big(mod(A)\big)$ is an abelian category.

I have a few questions about this category:

1. If $gl.dim(A)=0$, what is the global dimension of $Mor\big(mod(A)\big)$?
2. If $gl.dim(A)=1$, what structure does the category $Mor\big(mod(A)\big)$ has?
3. What can I say about $Mor\big(mod(A)\big)$ if $gl.dim(A)=n$, for a given integer $n$?
4. What if the global dimension of $A$ is infinite?