Induced monoidal structure from the functor

Abstract

Let $\mathcal{B}$ be a subcategory of a given category $\mathcal{D}$. Let $\mathcal{B}$ has monoidal structure. In this article, we discuss when can one extend the monoidal structure of $\mathcal{B}$ to $\mathcal{D}$ such that $\mathcal{B}$ becomes a sub monoidal category of monoidal category $\mathcal{D}$. Examples are discussed, and in particular, in an example of loop space, we elaborated all results discussed in this article.