This is Theorem 10.1.1 of Lind & Marcus's book, An Introduction to Symbolic Dynamics and Coding. They say that is "straightfordward" to go from

Let $X$ a shift of finite type and $Y$ a mixing shift of finite type such that $\text{Per}(X)\hookrightarrow\text{Per}(Y)$ and $h(X)<h(Y)$. Then, $X\hookrightarrow Y$.

to

Let $X$ and $Y$ irreducible shift of finite type such that $\text{Per}(X)\hookrightarrow\text{Per}(Y)$ and $h(X)<h(Y)$. Then, $X\hookrightarrow Y$.

How we can drop the mixing hypothesis on $Y$? I have thought in this all new year!