What should be the IUPAC name of $\ce{(NH3)5Co-NH2-Co(NH3)4(H2O)]}$?
This is similar to $\ce{[(NH3)5Co-NH2-Co(NH3)5](NO3)5}$, i.e. μ-amido-bis[pentaamminecobalt(III)] nitrate. But the two parts are different.
What should be the IUPAC name of $\ce{(NH3)5Co-NH2-Co(NH3)4(H2O)]}$?
This is similar to $\ce{[(NH3)5Co-NH2-Co(NH3)5](NO3)5}$, i.e. μ-amido-bis[pentaamminecobalt(III)] nitrate. But the two parts are different.
I shall respond to this question with all the rules to be followed when there is a compound in the given formats:
I hope you are familiar with the basic rules to be followed (ligands in alphabetical order, oxidation number written at the end in Roman numerals, atom of the ligand bonding with metal to be written in the case of ambidentate ligands, etc...). You can look up here - I have linked a Chemistry LibreTexts document.
Rule 1: The ligands in the middle should be written with the Greek letter $\mu_n$ prefixed to it - where $n$ indicates the number of atoms to which the ligand is bonded. In practice, the subscript $n$ is omitted when $n=2$. For ease of reference, we'll call this part of the name Part 1.
Rule 2: The metal atoms must be named separately - $\ce{L_1M_1}$ and $\ce{M_2L_2}$. Call this part of the name as Part 2 and Part 3 respectively. The OP has named an example - $\ce{[(NH3)5Co−NH2−Co(NH3)5](NO3)5}$ is called μ-amido-bis[pentaamminecobalt(III)] nitrate.
Rule 3: If metal + ligand part is repetitive, use prefix bis-, tris-, tetrakis- etc.
Oxidation numbers should be included separately, and for an individual metal, it is calculated by taking net negative charged ligands around it.
The final name would be in the format $\mathbf{\text{Part 2 Part 1 Part 3}}$.
The compound in your contention, $\ce{[\color{red}{(NH3)5Co}-\color{blue}{NH2}-\color{magenta}{Co(NH3)4(H2O)}]}$ : I have divided it into 3 parts (refer to the colour scheme).
The $\ce{\color{red}{red part}}$ is named pentaamminecobalt(III).
The $\ce{\color{blue}{blue part}}$ is named μ-amido.
The $\ce{\color{magenta}{pink part}}$ is named tetraammineaquacobalt(III).
Merging all the parts together:
pentaamminecobalt(III)-μ-amido-tetraammineaquacobalt(III).
Another example for your understanding:
is named as
Tetraamminechromium(III)-μ-amido-μ-hydroxidobis(ethylenediamine)cobalt(III) ion.
Sources: