If we have a category $\mathcal{C}$, then we can see it as an $\infty$-category. Furthermore, we can truncate and $\infty$-category $\mathcal{X}$ to get a category $\mathcal{X}_{\leq 1}$. My question is if these functors are adjoint, i.e. if we have $$\text{Hom}_{\mathfrak{Cat}}(\mathcal{X}_{\leq 1}, \mathcal{Y})\cong \text{Hom}_{\infty\text{-}\mathfrak{Cat}}(\mathcal{X},\mathcal{Y})?$$

If we have a category $\mathcal{C}$, then we can see it as an $\infty$-category. Furthermore, we can truncate and $\infty$-category $\mathcal{X}$ to get a category $\mathcal{X}_{\leq 1}$. My question is if these functors are adjoint, i.e. if we have $$\text{Hom}_{\mathfrak{Cat}}(\mathcal{X}_{\leq 1}, \mathcal{Y})\cong \text{Hom}_{\infty\text{-}\mathfrak{Cat}}(\mathcal{X},\mathcal{Y})$$ where $\mathcal{Y}$ is a category (which on the right is seen as an $\infty$-category).