On a lecture on tensors by XylyXylyX the concept of the "universal property" is introduced.
And there is a notational problem I am encountering:
So far, the notation for a tensor product in these lectures was, for example:
$$\underbrace{A^{\alpha\beta\gamma}{}_{\mu\nu\delta}}_{\text{components}}\, \underbrace {e_\alpha\otimes e_\beta\otimes e_\gamma}_{\text{basis vecs }\in V}\otimes \underbrace{e^\mu\otimes e^\nu\otimes e^\delta}_{\text{covecs }\in V^*}\tag 1$$
at times simply abbreviated as $A^{\alpha\beta\gamma}{}_{\mu\nu\delta},$ and making reference to a $(3,3)$-tensor product space (3 vectors / 3 covectors).
Now, on this lecture, there is some confusion in the notation, as reflected on the comments, so I'm trying to not make bad assumptions interpreting the expression:
$$C\left(v_1 \otimes v_2 \otimes v_3\otimes w^1 \otimes w^2 \otimes w^3\right)\tag 2$$
Is the expression inside the parenthesis $C(\cdot)$ a product tensor of the form in Eq.$(1)?$ with $v_1 =A^\alpha\,e_\alpha$ and $w^1 =A_\mu e^\mu?$
and
what is the role of $C?$
...so far the idea was to apply these tensor products to, in my example in Eq $(1)$, $3$ covectors and $3$ vectors, mapping to $\mathbb R,$ i.e. "it eats three covectors and three vectors."
For example, if I clipped out the $\delta$ in Eq.$(1)$ for clarity, the expected operation would be
$$T^{\alpha\beta\gamma}{}_{\mu\nu}\, \Big [ e_\alpha \otimes e_\beta \otimes e_\gamma\otimes e^\mu\otimes e^\nu\Big ]\left(\underbrace{B_\eta e^\eta, C_\omega e^\omega, F_\epsilon e^\epsilon}_{\text{eats 3 covectors}},\underbrace{Z^\theta e_\theta, Y^\rho e_\rho}_{\text{eats 2 vectors}}\right)\to \mathbb R$$
what is he doing now with $C(\cdot)?$
I see that $C(\cdot)$ is probably meant to modify the tensor product $v_1 \otimes v_2 \otimes v_3\otimes w^1 \otimes w^2 \otimes w^3$ getting rid of a vector and a covector by applying the dual space mapping, $\langle w^2,v^2\rangle$, but I wanted to confirm all the assumptions, and get perhaps a reformulation of this universal property.
Here is the slide:
What is the difference between $\small Z(v_1,v_2,v_3,w^1,w^2,w^3)$ and $\small C(v_1 \otimes v_2 \otimes v_3 \otimes w^1 \otimes w^2 \otimes w^3)?$