さてと・・・それじゃあ捜査を始めるとしようか。


Here's the sentence :

> さてと・・・それじゃあ捜査を始めるとしようか。

I would have understood if the sentence was : "さてと・・・それじゃあ捜査を始めよう", but I
don't understand the rest of the sentence. Is "としよう" some kind of grammar ?

As for the context, the characters were just about to investigate a murder.




捜査を始めよう means "Let's start the investigation", but 捜査を始めるとしようか means something
more along the lines of "should we try and start the investigation",
basically. It's along the lines of ～ようとする・～ようにする

食べようとする I'm about to eat. (I'm going to try to eat) (Maybe you don't have a
lot of time or something.)(attempt to do something)

食べるようにする I will try to eat. (Try to eat something you don't like, or maybe
you're sick)(non-momentum, or continuous conscious effort)

