In general, I do not think the statement in the title is true. But from Galois Theories, by Francis Borceus and George Janelidze, they claimed a similar fact without proof. I shall give sufficient background for the question before quoting his words.
Let $K\subset L$ be a Galois extension and let $\operatorname{Split}_K(L)_f$ be the category of all finite-dimensional $F$-algebras $A$ splitted by $L$, that is, every minimal polynomial over $K$ of an element in $A$ splits into distinct linear factors in $L$. Let $\operatorname{Aut}_K(L)\text{-Set}_f$ be the category of all finite $\operatorname{Aut}_K(L)$-sets. The authors proved that the Hom functor $$\operatorname{Hom}_K(-,L):\operatorname{Split}_K(L)_f\to\operatorname{Aut}_K(L)\text{-Set}_f$$mapping an $K$-algebra to the $\operatorname{Aut}_K(L)$-set $\operatorname{Hom}(A,L)$ of all $F$-morphisms from $A$ to $L$ is full and faithful. This is the theorem statement on page 28.
On page 34, the authors stated
Consider two algebras $A,B$ in $\operatorname{Split}_K(L)_f$. Composing with the projections $$A\twoheadleftarrow A\times B\twoheadrightarrow B$$yields maps $$\operatorname{Hom}_K(A,L)\hookrightarrow\operatorname{Hom}_K(A\times B,L)\hookleftarrow\operatorname{Hom}_K(B,L)$$which are injective since the projections are surjective."
Question. How come the surjectiveness of the projections implies the injectiveness of the morphisms induced by them? Is there anything I am missing from the property of this $K$-morphism Hom functor?
On page 34 again, after he showed the functor $\operatorname{Hom}_K(-,L)$ is a categorical equivalence, he said,
To conclude this chapter, it remains to observe that the Galois theorem we have just proved contains the classical Galois theorem. In deed, the contravariant equivalence of theorem 2.4.3 implies in particular the existence of an isomorphism between the lattice of subobjects $M$ $$K\hookrightarrow M\hookrightarrow L$$in $\operatorname{Split}_K(L)_f$ and the lattice of quotients $\operatorname{Hom}_K(K,L)$ $$\operatorname{Aut}_K(L)\simeq\operatorname{Hom}_K(L,L)\twoheadrightarrow\operatorname{Hom}_K(M,L)\twoheadrightarrow\operatorname{Hom}_K(K,L)\simeq\{\ast\}$$ in $\operatorname{Aut}_K(L)$-Set$_f$.
Similar Question. But how come the injections $$K\hookrightarrow M\hookrightarrow L$$induce surjections $$\operatorname{Hom}_K(L,L)\twoheadrightarrow\operatorname{Hom}_K(M,L)\twoheadrightarrow\operatorname{Hom}_K(K,L)?$$He is clearly NOT using the epi and mono definition for injection and surjection here.
Add. I think I have an idea for the first question. The Hom functor $\operatorname{ModHom}_K(-,M)$ of $K$-modules transforms surjection to injection since $\operatorname{ModHom}_K(N,-)$ is left exact. Thus for a surjection of two $K$-algebras $A\twoheadrightarrow B$ one has an injection $$\operatorname{ModHom}_K(B,L)\hookrightarrow\operatorname{ModHom}_K(A,L)$$ and clearly the restriction to the $K$-algebra Hom set $\operatorname{Hom}_K(B,L)$ is exactly the induced morphism $$\operatorname{Hom}_K(B,L)\to \operatorname{Hom}_K(A,L).$$
Being surjective (i.e. an epimorphism) in a category is defined as $\hom(f, X)$ being injective (as maps of sets) for all $X$. If you learned a different definition of epimorphism, stare at it for a few minutes, and you will see that you actually learned the same definition.