Consider the following example:
$C$ is a category each of whose Hom-sets is partially ordered. Let $f$, $g$, and $h$ are morphisms of this category. Consider the formula: $g\circ f \ge h$.
Intuition suggests that from this formula it follows:
- $f$ and $g$ are composable that is the destination of $f$ is the source of $g$.
- $g\circ f$ and $h$ lie in the same Hom-set, that is the source of $f$ is the same as the source of $h$ and destination of $g$ is the same as the destination of $h$.
So, intuition suggests that these conditions 1 and 2 follow from the formula. But can we define it formally, in order not to say explicitly that these equalities (1 and 2) of sources and destination hold? It is very boring to formulate such conditions explicitly in every statement about morphisms in a category.