https://codesandbox.io/s/nn53n493km

ReactCarousel.jsx

import React from 'react';
import { render } from 'react-dom';
import './style.css';

const cats = [
  "http://www.pawculture.com/uploads/small-cat-breeds-card.jpg",
  "http://www.pawculture.com/uploads/cats-small-places-card.jpg",
  "http://facts.net/wp-content/uploads/2015/07/Cat-Facts.jpg"
]

class App extends React.Component {
  constructor() {
    super();
    this.state = {
      cur: 0
    };
  }

  handleToTheLeft = () => {
    this.setState(function (state, props) {
      return {
        cur: this.state.cur ? this.state.cur - 1 : cats.length - 1,
      }
    });
  }

  handleToTheRight = () => {
    this.setState(function (state, props) {
      return {
        cur: (this.state.cur !== cats.length - 1) 
          ? this.state.cur + 1 
          : 0,
      }
    });
  }

  render() {
    return (
      <div>
        <button onClick={this.handleToTheLeft}>&lt;</button>  
        <img src={cats[this.state.cur]} alt="cat" />
        <button onClick={this.handleToTheRight}>&gt;</button>          
      </div>
    );
  }
}

render(<App />, document.getElementById('root'));