# On the definition of a 2-category

Wikipedia begins the list of ingredients in the definition of a 2-category as follows:

• A class of 0-cells (or objects) $A$, $B$, ....
• For all objects $A$ and $B$, a category $\mathbf{C}(A,B)$. The objects $f,g:A\to B$ of this category are called 1-cells and its morphisms $\alpha:f\Rightarrow g$ are called 2-cells; the composition in this category is usually written $\circ$ or $\circ_1$ and called vertical composition or composition along a 1-cell.

What is the reason for requiring $f$ and $g$ to be morphisms between the same pair of objects? Would it be possible/meaningful to generalize the definition a little and allow $f$ and $g$ to be between different pairs of objects: $f:A\to B$, $g:C\to D$?

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Yes, $2$-categories (to be precise: bicategories) arise quite naturally (math.stackexchange.com/questions/148134). And of course they are easier to handle than double categories. – Martin Brandenburg Aug 20 '13 at 16:16