Example 1

The composite of two strict mappings is not necessarily strict. In fact, every continuous mapping $f: X\rightarrow Y$ is the composite of the mapping pr$_2: X\times Y\rightarrow$ Y and of the mapping $x\mapsto (x, f(x))$ of X into $X\times Y$, which are both strict (TG, I, p. 26, prop. 5 and TG, I, p. 25, cor. 2). On the other hand, the composite of two strict and injective (resp. surjective) mappings is a strict mapping.