The identity natural transformation $\text{id}_{F}\colon F\Rightarrow F$ of $F$ is the natural transformation consisting of the collection
\[ \webleft\{ \webleft (\text{id}_{F}\webright )_{A}\colon F\webleft (A\webright )\to F\webleft (A\webright )\webright\} _{A\in \text{Obj}\webleft (\mathcal{C}\webright )} \]
defined by
\[ \webleft (\text{id}_{F}\webright )_{A}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{id}_{F\webleft (A\webright )} \]
for each $A\in \text{Obj}\webleft (\mathcal{C}\webright )$.
The naturality condition for $\text{id}_{F}$ is the requirement that, for each morphism $f\colon A\to B$ of $\mathcal{C}$, the diagram
commutes. This follows from unitality of the composition of $\mathcal{D}$, as we have
\begin{align*} F\webleft (f\webright )\circ \text{id}_{F\webleft (A\webright )} & = F\webleft (f\webright )\\ & = \text{id}_{F\webleft (B\webright )}\circ F\webleft (f\webright ),\\ \end{align*}
where we have applied unitality twice.
Let $A$ and $B$ be monoids and let $f,g\colon A\rightrightarrows B$ be morphisms of monoids. Applying the delooping construction of 
, we obtain functors $\mathsf{B}{f},\mathsf{B}{g}\colon \mathsf{B}{A}\rightrightarrows \mathsf{B}{B}$. We then have
\[ \text{Nat}\webleft (\mathsf{B}{f},\mathsf{B}{g}\webright )\cong \webleft\{ b\in B\ \middle |\ \begin{aligned} & \text{for each $a\in A$, we}\\ & \text{have $bf\webleft (a\webright )=g\webleft (a\webright )b$}\end{aligned} \webright\} . \]
Unwinding the definitions in this case, we see that a transformation $\alpha $ from $\mathsf{B}{f}$ to $\mathsf{B}{g}$ consists of a collection
\[ \webleft\{ \alpha _{\bullet }\colon \bullet \to \bullet \webright\} _{\bullet \in \text{Obj}\webleft (\mathsf{B}{A}\webright )} \]
of morphisms of $\mathsf{B}{B}$ indexed by $\text{Obj}\webleft (\mathsf{B}{A}\webright )$. Since $\text{Obj}\webleft (\mathsf{B}{A}\webright )=\text{pt}$ and the morphisms of $\mathsf{B}{B}$ are precisely the elements of $B$, it follows that $\alpha $ corresponds precisely to the data of an element $b\in B$. Now, a transformation $\webleft [b\webright ]\colon \mathsf{B}{f}\Rightarrow \mathsf{B}{g}$ is natural precisely if, for each $a\in \textup{Hom}_{\mathsf{B}{A}}\webleft (\bullet ,\bullet \webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}A$, the diagram
commutes. Unwinding the definitions, we see that this diagram is given by
and hence corresponds precisely to the condition $g\webleft (a\webright )b=bf\webleft (a\webright )$.