Let $k$ be a field and $A$ a $k$-algebra. Let $k \subseteq F$ be a field extension. Then naturally we have a restriction functor $\mathsf{res}: \operatorname{\mathit F-\mathbf{Alg}} \rightarrow \operatorname{\mathit k\,-\mathbf{Alg}}$. ($\operatorname{\mathit F-\mathbf{Alg}}$ means the algebras over field $F$).
I have seen that the functor $\mathsf{res}$ has the left adjoint $F \otimes_k-: \operatorname{\mathit k\,-\mathbf{Alg}} \rightarrow \operatorname{\mathit F-\mathbf{Alg}}$. The unit of this adjunction on $A$ is the $k$-algebra homomorphism $$\varphi: A \rightarrow F\otimes_kA$$ $$a\mapsto 1\otimes a$$ Also, $\varphi$ is injective.
I am not familiar with those things. So how to get that $\text{Hom}(F\otimes_k A_1,A_2) \cong \text{Hom}(A_1,\mathsf{res}(A_2))$ for any $k$-algebra $A_1$ and $F$-algebra $A_2$? Another question is that I think the unit of this adjunction is of the form $\varphi: A \rightarrow \mathsf{res}(F \otimes_k A)$, why it writes as $A \rightarrow F\otimes_k A$? Thank you for any help.