The notation quoted in the original post is flat out wrong.

The presence of the "$-$" in the notation $\hom(A,-)$ is meant to indicate where the argument to the functor is to be placed: when evaluated at some variable $X$ of type $\mathbf{A}$ (e.g. $X$ could be an object or an arrow of $\mathbf{A}$), one is supposed to write $\hom(A,X)$.

The notation $\hom(A,-)(X)$, however, indicates a function that, when evaluated at $Y$, produces the value $\hom(A,Y)(X)$, and that is definitely not what is intended (and is usually nonsensical!).

The notation $\hom(A,-)$ is itself notation for partially evaluating the functor $$\hom(-,-) : \mathbf{A}^\circ \times \mathbf{A} \to \mathbf{Set}$$ at $A$ in its first argument.

The notation quoted in the original post is flat out wrong.

The presence of the "$-$" in the notation $\hom(A,-)$ is meant to indicate where the argument to the functor is to be placed: when evaluated at some variable $X$ of type $\mathbf{A}$ (e.g. $X$ could be an object or an arrow of $\mathbf{A}$), one is supposed to write $\hom(A,X)$.

The notation $\hom(A,-)(X)$, however, indicates a function that, when evaluated at $Y$, produces the value $\hom(A,Y)(X)$, and that is definitely not what is intended (and is usually nonsensical!).

The notation $\hom(A,-)$ is itself notation for partially evaluating the functor $$\hom(-,-) : \mathbf{A}^\circ \times \mathbf{A} \to \mathbf{Set}$$ at $A$ in its first argument.

Alternative notations do exist, though. For example, the two notations

$$ h_Y(X) = h^X(Y) = \hom(X, Y) $$

get used. I have also seen $\mathbf{y}$ used for the Yoneda embedding $\mathbf{A} \to \mathbf{Set}^{\mathbf{A}^{\circ}}$; that is, $\mathbf{y}A = \hom(-,A)$, although this has the wrong variance for the specific example under discussion. I don't think I've seen the other embedding notated by the letter 'y' before.