Horatio was silent upon Jack's request. The promise to send an army out into war was not a simple request. War consumed resources. Not to mention there was also the risk of losing, which would put the country in a worse position than before the war.
After a while, Horatio said to Jack, "Very well. You have my word. I will help you defeat Liguritudum and its allies if I become sovereign."
Jack nodded. He preferred Horatio to take the time to answer him. This meant Horatio truly pondered the term. If Horatio answered immediately without pause, he would be suspicious instead. Suspicious that Horatio would just deny the promise after becoming sovereign.
There was still the chance of that, of course, but Horatio had held his end of the bargain when Jack delivered the Book of the Damned. Jack figured this showed Horatio was a man of his word. But most of all, Jack didn't truly have many options to begin with. He had to deal with Emperor Marcus one way or the other.
