# Confusion about tensor fields taking values in vector fields rather than functions

I'm beginning with tensorial calculus and I have some questions. Let $$(M,g)$$ a riemannian manifold with $$\nabla$$ his Levi Civita connection. The curvature tensor $$R$$ is defined as

\begin{align*} R : \mathfrak{X}(M) \times \mathfrak{X}(M) \times \mathfrak{X}(M) &\to \mathfrak{X}(M)\\ (X,Y,Z) &\mapsto R(X,Y)Z \end{align*}

where

$$R(X,Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X,Y]}Z.$$

In all books of differential geometry, they said that $$R$$ is a $$(1,3)$$ tensor but I don't know why, because a $$(1,3)$$ tensor is a multilinear map

$$\Omega^1(M) \times \mathfrak{X}(M) \times \mathfrak{X}(M)\times\mathfrak{X}(M) \to C^\infty(M).$$

I think that I don't understand any concept or because for example, let $$X \in \mathfrak{X}(M)$$, then $$\nabla X : TM \to TM$$, $$(\nabla X)(v_p) = \nabla_{v_p}X$$, is a $$(1,1)$$ tensor field and I don't know why.

$$\textbf{Remark}$$: $$\Omega^1(M)$$ is the set of all 1-forms, $$\alpha : M \to TM^*$$.

A $$(p, q)$$-tensor field on a smooth manifold $$M$$ is a $$C^{\infty}(M)$$-multilinear map $$T : \Omega^1(M)^p\times\mathfrak{X}(M)^q \to C^{\infty}(M)$$.
Given a $$C^{\infty}(M)$$-multilinear map $$S : \Omega^1(M)^p\times\mathfrak{X}(M)^q \to \mathfrak{X}(M)$$, there is an associated $$(p + 1, q)$$-tensor field $$T : \Omega^1(M)^{p+1}\times\mathfrak{X}(M)^q \to C^{\infty}(M)$$ defined by
$$T(\beta, \alpha^1, \dots, \alpha^p, X_1, \dots, X_q) := \beta(S(\alpha^1, \dots, \alpha^p, X_1, \dots, X_q)).$$
Likewise, given a $$C^{\infty}(M)$$-multilinear map $$S : \Omega^1(M)^p\times\mathfrak{X}(M)^q \to \Omega^1(M)$$, there is an associated $$(p, q + 1)$$-tensor field $$T : \Omega^1(M)^p\times\mathfrak{X}(M)^{q+1} \to C^{\infty}(M)$$ defined by
$$T(\alpha^1, \dots, \alpha^p, Y, X_1, \dots, X_q) := (S(\alpha^1, \dots, \alpha^p, X_1, \dots, X_q))(Y).$$