  It is shown that for a map $f \colon X \to Y$ of compact spaces the unital
$\ast$-homomorphism $C(f) \colon C(Y) \to C(X)$ is projective in the category
$\operatorname{Mor}({\mathcal C}^{1})$ precisely when $X$ is a dendrite and $f$
is either homeomorphism or a constant.
