@Eric, I am sorry but I still do not understand your second answer vis a vis my silly example. I do understand that I am allowed to vary one endpoint (i.e., one morphism). But it still has to be homotopic to the given map, right ? Then, it seems, all I have to choose from is the same map. Am I allowed to vary the complex itself ? Can you point out a reference where some examples of higher homotopies are worked out. Thanks !

Thanks, you are right - what follows from Joyal's theorem is the contractibility of the space of maps equipped with a homotopy to a given map. Still... does not it imply that for the complex above all the homotopies should be equal ? And they are not ?