$\textbf{4.1.} \enspace \triangleright$ Composition is defined for two morphisms. If more than two morphisms are given, e.g.,

then one can compose them in several ways, for example:

$$(ih)gf, \quad (i(hg))f, \quad i((hg)f), \quad \text{etc.}$$

so that at every step one is only composing two morphisms. Prove that the result of any such nested composition is independent of the placement of the parentheses.
(Hint: Use induction on $n$ to show that any such choice for $f_nf_{n-1}\cdots f_{1}$ equals

$$((\cdots((f_nf_{n-1})f_{n-2})\cdots)f_1).$$

Carefully working out the case $n=5$ is helpful.) $[\S4.1, \S\textrm{II}.1.3]$

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$\textbf{4.2.} \enspace \triangleright$ In Example 3.3 we have seen how to construct a category from a set endowed with a relation, provided this latter is reflexive and transitive. For what types of relations is the corresponding category a groupoid (cf. Example 4.6)? $[\S 4.1]$