There are some exercises where I'm not fully sure that it's necessary to prove both directions. I'm using the below exercise as one example, though I'm not, per se, confused on how to solve this. This is exercise 2.1.1 in Artin's algebra textbook.
Let $S$ be a set. Prove that the law of composition defined by $ab = a$ for all $a$ and $b$ in $S$ is associative. For which sets does this law have an identity?
The second part of the problem begs the question of whether this requires a two-sided proof. First I assume that $S$ has an identity $e$ and it isn't hard to show that $S = \{e\}$. Do I need to prove the "converse," though, that if $S$ is a singleton, then this single element is an identity?
This seems rather trivial, but I'm not totally sure what qualifies as fully rigorous here. In the first step, I haven't "proved" the existence of an identity, but said only that if it has one, that's the entirety of $S$. I'm actually, in some sense, "establishing" an identity in the reverse direction, though there isn't much verification to it. If $S = \{a\}$, then $aa = a$ and $a$ is an identity.
If, in the first step, you only proved that if $S$ has an identity element $e$, then $S=\{e\}$, then something is missing. Namely: the fact that $\{e\}$ endowed such an operation has an identity element. Yes, it is true, and it is easy to prove.
That's all. The problem does not require you to prove that if a set $S$ is a singleton, then it is possible to define on $S$ an operation with the required property (although this is true and easy to prove).