When considering the boundary and coboundary maps, we have the common definitions that the boundary map based on the space of chains $C_k(X)$ is $$\partial_k([v_0,...,v_k])=\sum_{i=0}^k (-1)^i[v_0,...,v_{i-1},v_{i+1},...,v_k],$$ and the coboundary map $\delta_k$ based on the space of cochains $C^k(X)$ is the adjoint operator of $\partial_{k+1}$, i.e. $$(\delta_k f)([v_0,...,v_{k+1}])=f(\partial_{k+1}[v_0,...,v_{k+1}]) =\sum_{i=0}^k (-1)^if([v_0,...,v_{i-1},v_{i+1},...,v_{k+1}])$$

See https://magnus.ece.gatech.edu/Papers/MuhammadEgerstedtMTNS06.pdf page 4.

In the paper, the authors define the $k$-combinatorial Laplacians $\Delta_k:C^k(X;\mathbb{R})\to C^k(X;\mathbb{R})$ and $\mathcal{L}_k:C_k(X;\mathbb{R})\to C_k(X;\mathbb{R})$ by $$\Delta_k=\delta_{k-1} \delta^*_{k-1}+\delta^*_k\delta_k $$ $$\mathcal{L}_k=\partial_{k+1} \partial^*_{k+1}+\partial^*_k\partial_k $$

Here is my question:

For me, neither of these two $k$-combinatiorial Laplacians seem well defined. For example, $\delta_k$ works on a cochain $f\in C^k(X;\mathbb{R})$ and after that we get a cochain in $C^{k+1}(X;\mathbb{R})$, but how can we proceed with $\delta^*_k$? Since $\delta^*_k = \partial_{k+1}$, how can we put $\partial_{k+1}$ on a cochain?

Can you give me an example of the $\delta^*_k\delta_k$ on any cochain $f$?

For instance, what is, $\delta^*_1\delta_1 f([v_0,v_1])= \partial_2\delta_1 f([v_0,v_1])$?