Vtyjkyuk

I am having some difficulty understanding a piece of notation from Riemannian Geometry: and Introduction to Curvature by John M. Lee.

In Section 2 just under equation 2.3 Lee defines the trace operator which lowers the rank of a tensor by 2.

He defines the map:
$$mathrmtr:T_l+1^k+1(V)longrightarrow T_l^k(V)$$
By letting:
$$mathrmtr; F(omega^1,dotsomega^l,V_1,dots,V_k)$$
be the trace of the endomorphism:
$$F(omega^1,dots,omega^l,bullet,V_1,dots,V_k,bullet)$$

But how is it that $F(omega^1,dots,omega^l,bullet,V_1,dots,V_k,bullet) inmathrmEnd(V)$, it looks like it should belong to $T_l+1^k+1$, I think my confusion lies with the $bullet$ in the above expression. Unforturnately I cannot find any explanation of this notation in the textbook. Is this notation common for something that I am not aware of?

For fixed $omega^1, ldots, omega^l in V^*$ and $V_1, ldots, V_k in V$ the notation $F(omega^1,dots,omega^l,bullet,V_1,dots,V_k,bullet)$ signifies an element $G in T_1^1(V)$ such that
$$G(omega^l+1, V_k+1) = F(omega^1, ldots, omega^l, omega^l+1, V_1, ldots, V_k, V_k+1).$$

Then $operatornametr F in T_l^k(V)$ is defined by
$$(operatornametr F)(omega^1, ldots, omega^l, V_1, ldots, V_k) = operatornametrG.$$