Labeled commutative diagram Consider a commutative diagram.
For example the following diagram in $\mathbf{Set}$:
$$
\begin{array}{ccc}
& \overset{+1}{\longrightarrow} &\\
\mathbb{Z} & & \mathbb{Z} \\
& \overset{-1}{\longleftarrow} &
\end{array}
$$
In this diagram a morphism between two objects is uniquely determined by two vertices of the diagram. But it is not uniquely determined by a pair of objects, because the same object corresponds to several different vertices.
I want a concept of "labeled" commutative diagram, in order that I could determine (unique by a condition I require) the morphism by a pair of vertices.
Are such "labeled commutative diagrams" (specifically with the case if the morphism is uniquely determined by a pair of vertices) a known concept? How is it named? Where to look about this?

A diagram in category $A$ is most commonly defined as a functor from a small category $D$ to $A$.
In this view, we can say, the diagrams are automatically labeled by objects and morphisms of $D$.
For your specific example, take the free category $D$ on the graph with two (different) objects and two arrows as in your picture $1\overset{a}{\underset{b}\rightleftarrows}2$. 
Then your diagram becomes a functor $D\to\mathcal{Set}$, and you can refer to either vertex $\Bbb Z$ as $D(1)$ or $D(2)$.
Identifying an arrow by a pair of vertices in the diagram is possible whenever the graph we start with has no parallel edges.
