Let $X,Y$ be $\mathbb C$-Hilbert spaces and $X\otimes Y$ denote the completion of the Hilbert space tensor product of $X$ and $Y$.
Let $y\in Y$ and $A_y:X\to X\otimes Y\to X$ denote the unique linearization of $$x\otimes y'\mapsto\langle y,y'\rangle_Yx.\tag1$$
If $u\in X\otimes Y$, then we should have $$\langle x,A_yu\rangle_X=\langle x\otimes y,u\rangle_{X\otimes Y}\tag2.$$
Question: Is there a name and common notation for the operator $A_y$? It seems to be some kind of projection (in light of $(1)$) and is "compatible" with $X\otimes Y$ (in light of $(2)$).
The context in which this question came to mind is the definition of partial traces, vector-valued traces and tensor contractions.
I've seen the notation ${}_{X}\hspace{-.25em}\left\langle y\right|$, but I think this is really hard to read. For example the left-hand side of $(2)$ then reads $\langle x,{}_{X}\hspace{-.25em}\left\langle y\right|u\rangle_X$ ...
Moreover, I think it is likely to be confused with the "bra" $\left\langle y\right|:=\langle y,\;\cdot\;\rangle_Y$ from the bra-ket notation (which I also don't like, but since it is quite popular ...)