We consider the category $\mathcal{K}$ with an element $a$ and $\text{Hom}(a,a) = S$, where $S$ is a semigroup with unit. Is $a$ an injective object of $\mathcal{K}$, if $S = \{ 1,\alpha, \alpha^2\}$ is a semigroup with 3 elements and $\alpha^3 = \alpha^2$?
Firstly, we want to study the monomorphisms in category $\mathcal{K}$.
It is trivial, that $\varphi=1$ is a monomorphism. We consider $\varphi = \alpha$, then $\alpha \cdot \alpha = \alpha^2 \cdot \alpha$, by assumption, but $\alpha \neq \alpha^2,$ so $\varphi$ is not a monomorphism.Now, let $\varphi = \alpha^2$, using a similar proof, we obtain that $\varphi$ is not monomorphism.
Summarize, the category $\mathcal{K}$ has only one monomorphism.
Then for $\varphi = 1$ and $\forall \beta \in S $ there exists $\gamma = \beta \in S$ such that $\gamma \cdot \varphi = \beta$, so $a$ is an injective object.
Is this solution correct?