Is tensor product of two global sections commutative in the tensor product sheaf?

Question

Let $\mathcal{L}$ be an invertible sheaf on an algebraic curve $X$. Since $\mathcal{L}\otimes \mathcal{L}$ is the sheafification of the presheaf $(\mathcal{L}\otimes \mathcal{L})^{pre}$, we have a morphism $$ \varphi:H^0(\mathcal{L})\otimes H^0(\mathcal{L})\rightarrow H^0(\mathcal{L}\otimes \mathcal{L}). $$ Let $s,t\in H^0(\mathcal{L})$. We do not necessarily have that $s\otimes t=t\otimes s$.

Does the equality $\varphi(s\otimes t)=\varphi(t\otimes s)$ hold?